code: def ellipse_to_laf(ells: Tensor) -> Tensor:
      """Convert ellipse regions to LAF format.
  
      Ellipse (a, b, c) and upright covariance matrix [a11 a12; 0 a22] are connected
      by inverse matrix square root: A = invsqrt([a b; b c]).
  
      See also https://github.com/vlfeat/vlfeat/blob/master/toolbox/sift/vl_frame2oell.m
  
      Args:
          ells: tensor :math:`(B, N, 5)` of ellipses in Oxford format [x y a b c].
  
      Returns:
          LAF :math:`(B, N, 2, 3)`
  
      Example:
          >>> input = torch.ones(1, 10, 5)  # BxNx5
          >>> output = ellipse_to_laf(input)  #  BxNx2x3
      """
      KORNIA_CHECK_SHAPE(ells, ["B", "N", "5"])
      B, N, _ = ells.shape
      # Previous implementation was incorrectly using Cholesky decomp as matrix sqrt
      # ell_shape = concatenate([concatenate([ells[..., 2:3], ells[..., 3:4]], dim=2).unsqueeze(2),
      #                       concatenate([ells[..., 3:4], ells[..., 4:5]], dim=2).unsqueeze(2)], dim=2).view(-1, 2, 2)
      # out = torch.matrix_power(torch.cholesky(ell_shape, False), -1).view(B, N, 2, 2)
  
      # We will calculate 2x2 matrix square root via special case formula
      # https://en.wikipedia.org/wiki/Square_root_of_a_matrix
      # "The Cholesky factorization provides another particular example of square root
      #  which should not be confused with the unique non-negative square root."
      # https://en.wikipedia.org/wiki/Square_root_of_a_2_by_2_matrix
      # M = (A 0; C D)
      # R = (sqrt(A) 0; C / (sqrt(A)+sqrt(D)) sqrt(D))
      a11 = ells[..., 2:3].abs().sqrt()
      a12 = torch.zeros_like(a11)
      a22 = ells[..., 4:5].abs().sqrt()
      a21 = ells[..., 3:4] / (a11 + a22).clamp(1e-9)
      A = stack([a11, a12, a21, a22], dim=-1).view(B, N, 2, 2).inverse()
      out = concatenate([A, ells[..., :2].view(B, N, 2, 1)], dim=3)
      return out
  
imports: import math
  from typing import List, Optional, Tuple, Union
  import torch
  import torch.nn.functional
  from kornia.core import Tensor, concatenate, cos, sin, stack, tensor, zeros
  from kornia.core.check import KORNIA_CHECK_LAF, KORNIA_CHECK_SHAPE
  from kornia.geometry.conversions import angle_to_rotation_matrix, convert_points_from_homogeneous, rad2deg
  from kornia.geometry.linalg import transform_points
  from kornia.geometry.transform import pyrdown
label_desc: Use image node_id 9 for calling a global function with example usage: ellipse_to_laf(ells) and returns: out